On the Oscillation of Certain Third Order Linear Differential Equations

On the Oscillation of Certain Third-order Difference Equations

(ii) {g(n)} is a nondecreasing sequence, and limn→∞ g(n)=∞; (iii) f ∈ (R,R), x f (x) > 0, and f ′(x)≥ 0 for x = 0; (iv) αi, i= 1,2, are quotients of positive odd integers. The domain (L3) of L3 is defined to be the set of all sequences {x(n)}, n ≥ n0 ≥ 0 such that {Ljx(n)}, 0≤ j ≤ 3 exist for n≥ n0. A nontrivial solution {x(n)} of (1.1;δ) is called nonoscillatory if it is either eventually posi...

Oscillation of Third-order Functional Differential Equations

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear delay differential equation (E) ˆ a(t) ˆ x ′′(t) ̃ γ ̃ ′ + q(t)f(x [τ (t)]) = 0. Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of (E). Obtained results essentially improve and complement earlier ones. Variou...

Nonlinear oscillation of certain third-order neutral differential equation with distributed delay

The authors obtain necessary and sufficient conditions for the existence of oscillatory solutions with a specified asymptotic behavior of solutions to a nonlinear neutral differential equation with distributed delay of third order. We give new theorems which ensure that every solution to be either oscillatory or converges to zero asymptotically. Examples dwelling upon the importance of applicab...

ON TE EXISTENCE OF PERIODIC SOLUTION FOR CERTAIN NONLINEAR THIRD ORDER DIFFERENTIAL EQUATIONS

On the oscillation of certain third order nonlinear functional differential equations

In this paper we shall investigate the oscillatory properties of the equations d2 dt2 ( 1 a(t) ( dx(t) dt )α) + q(t)f (x[g(t)]) = 0 and d2 dt2 ( 1 a(t) ( dx(t) dt )α) = q(t)f (x[g(t)])+ p(t)h(x[σ(t)]), where α is the ratio of two positive odd integers. AMS subject classification: 34C10.